On Bott-Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms
Lorenzo Sillari, Adriano Tomassini
TL;DR
This work extends Bott-Chern and Aeppli cohomologies to almost complex manifolds by formulating $H^k_{d+d^c}$ and $H^k_{dd^c}$ from the operators $d$ and $d^c$, and clarifies their connections to de Rham, $d^c$, and (delta, bar-delta) cohomologies. It develops a comprehensive Hodge theory for $d$ and $d^c$, introducing four elliptic Laplacians whose harmonic spaces are finite-dimensional and relate to the cohomologies, yielding almost complex invariants such as $h^1_{d+d^c}$ that can distinguish between structures with identical Nijenhuis-tensor rank. A parity-based $\mathbb{Z}_2$-splitting of Bott-Chern cohomology and a precise picture of the relations between harmonic forms and cohomology spaces are established, with strong results in the almost Kähler 4-manifold case (e.g., a decomposition of $\mathcal{H}^2_{d+d^c}$ and equality of certain harmonic spaces with Tseng–Yau constructions). The paper provides concrete examples (notably Kodaira-Thurston and Sol(3)×S^1) highlighting the differences between harmonic spaces and cohomology, including cases where $H^2_{d+d^c}$ is infinite-dimensional. Overall, the framework broadens the toolkit for studying almost complex and almost symplectic geometry and yields new invariants and links to symplectic cohomologies.
Abstract
In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott-Chern and Aeppli cohomologies defined using the operators $d$, $d^c$. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to $d$, $d^c$, showing their relation with Bott-Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott-Chern cohomology of $1$-forms is finite-dimensional on compact manifolds and provides an almost complex invariant $h^1_{d + d^c}$ that distinguishes between almost complex structures. On almost Kähler $4$-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.